~How to Solve Max/Min Problems~

These types of problems are also called, "mini/max", "optimization", "extreme value" problems. Basically, there are two approaches. If numbers are "nasty" or zeroes of certain types of equations can not be found by easy methods, the TI-83 or other computer software must be used. Let's take the first category:

1) The derivative method: Extreme values of functions (high/low points on the curve) will occur at one of 3 places:

a) where the first derivative = 0. (tangent line is horizontal) (most common)

b) At the end points of the interval when a domain is specified or implied (must check these)

c) Where the derivative fails to exist. (rare, but does happen sometimes) (i.e., y=x^2/3 at x=0) (cusp with vertical tangent, but function is continuous there) Take a look with your TI-83. Use x in [-1;1], y in [-1;1].

To find these points in a), take the derivative, set it = 0, then solve for x. (assuming the numbers are nice & you can solve that equation).

If not, the TI-83 is used. Don't forget, we are looking for points, (a,b), so pick up the y-value from the table after you code in the original function.

The y-values of these points must be compared to the others (if any) from parts b) & c). To get the end points, just sub in the x-values there & use the table.

Once you have the derivative, check & see if there are any x-values that make the derivative undefined (usually where the bottom is zero or a negative under an even root). If there is a defined y-value for the original function there, find it & compare it to the other points found. Most of the time, there will not be one. (example given above is an exception)

Usually, we are after absolute max/min over the entire domain, so, in most instances, you can disregard the local or relative max/min. Make sure you read the wording of the problem & see if a closed interval is stated. If not, assume you are after the absolute max/min for the entire curve. If you are asked to find ALL extreme values for the function, include relative along with absolute values of the function. If the word points is included, then your answers will include both x and y values for the function.

A sketch would consolidate your understanding, so look at the y-values & set a friendly window for viewing the curve. If you did things correctly, you'll see their locations.

2) The TI-83 method: In the event that the numbers are "nasty" or the derivative equation can not be solved easily, use this technique.

a) code in the equation for which you wish to find max/min points

b) usually, a specified domain for x is indicated, so, go to the table & insert values in small increments from left to right that are in the interval of the domain. Take a look at their y-values in the 2nd column. This will give you a good idea of how to set up your window. For the x values, just use values that include the domain.

c) Now you are ready to view the curve. Make any necessary adjustments to your window so that you can clearly view all the high & low points of your curve over the domain indicated.

d) press 2nd trace (Calc). Go to appropriate menu for Maximum (4) or Minimum (3). Press enter. The curve should reappear & you'll be prompted to give a left guess (x-value) around a high point (if you are after a max pt). To get an idea of values on the left side of this point, use trace & move the marker with the cursor along the curve & read the x-value at the bottom of your viewing window. After entering the left guess, you will be prompted for a right guess. Do similar steps. After you enter that, you will be prompted for a guess (any number between the left & right guesses). Enter that, & walla walla, you answer comes out at the bottom of the screen. It may not be exact, since "nasty" numbers include repeating decimals & irrationals, so round off may be necessary. Always check the directions of the problem. It should be stated there.

Read your problem carefully & give the answer(s) that the problem calls for. Sometimes, only the value of the independent variable is needed (i.e., what production level (x) produces the largest profit (y) ) Most of the time both are asked for in problem (max/min points) (answers are in the form (x,y)). Also, in most cases, a graph is called for in a problem, so read the directions carefully.